How mobility impacts the flow-level performance of wireless data systems

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How mobility impacts the flow-level performance of wireless data systems

Thomas Bonald, Sem Borst, Alexandre Proutière

To cite this version:

Thomas Bonald, Sem Borst, Alexandre Proutière. How mobility impacts the flow-level performance of wireless data systems. Infocom, 2004, Hong Kong, China. �10.1109/INFCOM.2004.1354597�. �hal- 01244815�

How Mobility Impacts the Flow-Level Performance of Wireless Data Systems

T. Bonald*. S . C . Borst', A. Proutiere"*

.dbstracr-The potential for exploiting rate variations to in- creasp the capacity of wireless systems by opportunistic schedul- ing has been extensively studied at packet level. In the present paper, we examine hon slower, mobility-induced rate variations impact performance at flow level accounting for the random number of flows sharing the transmission resource. We identify hvo limit regimes, termed /hid and qttmi-smimoy. where the rate variations occur on an infinitely fast and an infinitely slow time scale, respectively. Using stochzstic comparison techniques, we show that these limit redmes provide simple performance hounds that only depend on essily calculated load factors.

Additionally, we prove that for B broad clnss of fading pm- cesses, performance varies monotically with the s p e d of the rate variations. These results are illustrsted through numerical erperiments. showing that the fluid and quasi-stationary bounds are remarkably tight in certain usual C B S C S .

I. INTRODUCTIOX

Nest-generation wireless nehvotks are expected to support a wide variety of high-speed data applications. in addition to comzentiod voice sewices and current low-bandwidth data services such as short messaging. The integration of these heterogeneous applications on a common transmission infrastmcture raises similar challenges as in wireline integrated network% In wireless emironments, these issues are further exacehated by interference problems: intrinsically limited bandwidth and highly variable and unpredictable propagation characteristics. Specifically, the channel quality may vruy widely among spatially distributed users due to diskmce- related attennation In addition the channel conditions for a given user may vary dramatically over time because of fading effects.

Fading is an extremely comples physical phenomenon caused by the interaction between the propagation emironmen1 and user mobility. It emerges in diverse forms and tqpically spans a wide range of time scales. Fast fading arises because of multi-path propagation effects. and as the word suggests, occurs at a relatively high pace. Slow fading manifests ifself at a more macroscopic level as a result of distance-related atten- nation and scattering due to obstacles and terrain conditions.

and evolves over a longer time scale.

Wireless circuit-switched voice networks rely on power control mechanisms for adjusting the transmit power to combat fading and maintain a fixed transmission rate. Various data applications on the other hand. such as Gle transfers and Web browsing sessions, are less sensitive to packet-level delays.

*France Telacom R&D. France. barresponding aulhor.

{ t h o m a s . b o n a l d , u l e s a d ~ ~ . p ~ ~ ~ i ~ ~ ~ } ~ f ~ ~ ~ ~ l ~ l ~ ~ ~ m . ~ ~ m ICWI. The Netherlands, [email protected], alsq dliliatcd with Bell Lab-

oratories. Lucent Technologies

and do not have a stringent rate requirement. Such elastic applications are well-suited for rate control algorithm which dynamically adapt the transmission rate over time so as to match the fluctuations in channel quality. The resulting varia- tions in the transmission rates in fact open up the possibility of scheduling data transmissions to the various users when their channel conditions are relatively favorable. While fading is considered to have a predominantly adverse impact for voice connections, it thus provides the opportunity to achieve throughput gains for elastic data transfen.

The performance gains from opportunistic scheduling rely on the rates v m i n g snflicientl?- slowly to be tmcked with reasonable accnracy. but relatively rapidly compared to the delay tolerance of the users. High-frequency fading causes estimation and prediction problems. diminishing the scope for scheduling. Slow variations cannot be harnessed. or only at the expense of compromising the delay allowance of the users. For example. typical values of the time constant in the Proportional Fair algorithm for the CDMA IxEV-DO system [6]; [ I l l . [I61 are between 10 and 1000 slots of 1.67 ms. This ensnres that stawation effects cannot pcnist for csccssive periods. but it also implies tlmt slower variations are not csploited. In practice, relatively low-mobility scenarios tend to provide the greatest potential for scheduling gains.

While the performance of opportunistic scheduling algo- rithms has been thoroughly explored at packet level [2].

[91. 1141, [I!)], [23], [26], the impact of fading on flow- level performance has received remarkably little attention so far. In [SI, it was shown that when fading is relatively fast compared to flow dynamics. the system may in certain cases be representcd by a Processor-Shanng type model with a state- dependent service rate that accounts for the scheduling gains.

This model provides explicit formulas for the distribution of the number of active Rows and of the mean transfer delay. In particular. performance is insens;t;ilee, in the sense that these fonnulas only depend on the characteristics of the system through an easily computed 'load' factor. The notion of 'cell capacity', critical for mmensioning purposes. can then be defined independently of precise statistics of offered traffic In the present paper. we focus on the impact of mobility- induced fading that evolves on a slower time scale and manifests itself in the form of independent rate variations at How level. Due to these slower rate variations. the insensitivity propelt). is lost, and performance depends in some complicated fashion on detailed rate statistics and uaffic characteristics of the system rendering exact analysis virmally impossible.

Considering these complexities. we c o r n p m the performance

~71.

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of the system with that in two limit regimes, tennedfluid and quasi-stationay. obtained when rate variations have the same instantaneous statistics, but occur on an infinitely fast and an infinitely slow time scale. respectively. Using stochastic comparison techniques, we show that the fluid and quasi- stationan. regimes yield optimistic and conservative perfor- mance estimates, respectively. These estimates are very usefuL since pelformance in the limit regimes is insensitive, and only depends on appropriately deIined load factors, thus providing simple bounds that render 11% detailed statistical character- istics of the system largely &levant. Numerical experiments indicate that these bounds are surprisingly tight in many cases.

The above ordering results show that infinitely fast rate fluctuations yield best performance, while infinitely slow vari- ations produce worst performance. It is tempting to conjecture tllat performance improves monotonically as the fading pro- cess is speeded up. We demonstrate that this is indeed the case for a broad class or Malkov-type fading processes. It is worth observing that these results relate to a change in the time scale of the rate variations for given instantaneous m e statistics. As mentioned above, the actual transmission rates inay be reduced at higher fading frequencies because of estimation and prediction problems, so for a given system a change in the time scale will also affect the marginal rate distributions to some extent.

At a qualitative level, the finding that mobility-related rate variations improve performance resonates with the generic principle described earlier with respect to opportunistic scheduling. It also ties in with the observation in 1131 that mobility increases the capacity of ad hoc wireless networks.

In the present context however. the performance improvement does not rely on opportunistic scheduling. Instead. mformally stated. it arises from the fact that flow-level performance measures behave as convex functions of the rate processes.

The remainder of the paper is organiied as follows. In Section lI we present a detailed model description. In Sec- tion 111 we introduce the fluid and quasi-stationary regimes mentioned above. We establish a necessary and sufficient stability condition in Section IV. In Section V it is proved that the fluid and quasi-stationary regimes provide stochastic bounds for the performance of the actual system. For Markov- type fading processes. we demonstrate in Section VI that performance in fact monotonically varies with the time scale of rate fluctuations. In Section VI1 we discuss some numerical eqeriments performed to illusuate the analytical results. In Section W I we make some concluding r e m a h .

11. MODEL DESCRIPTION

Consider a single base station whose transmission power is time shared by a dynamic number of elastic flows. Each flow is represented as a 'fluid' data transfer with a variable rate tlwt depends on the channel quality and the number of competing flows. Packet-level dynamics are implicitly repre- sented through the way flows share the transmission resource.

as explained below. Each flow is characterized by its size (in bits) and its 'feasible' transmission rate that varies relatively slowly, due notably to user mobility.

Specifically, we consider an arbitmy number K of flow classes, each class corresponding to given statistical flow size and rate variation characteristics. Class-k flows anive as a Poisson process of rate Xk. We denote by 4; the s u e of the i- th aniving class-k flow and by Rk;(t) its feasible rate at time t_

corresponding to the actual transmission rate of this Bow if it were the only one in the system. (For notational convenience, we define R k , ( t ) for all values of t. Note however that the i-th class-k flow may not have anived yet or may already be completed at time t , in which case the value of R h ; ( t ) is of no significance.) We assume tlut 4; and R k ; ( t ) , i = 1 , 2 , . . ~ are i.i.d. copies of a random variable Fk and a stationary and ergodic process Rk(t), respectively. The process Rk(t) is assumed to be bounded and right-continuous with left-band limits.

Let Ck = E I R k ( 0 ) ] be the time-average feasible rate of a class-k flow. We define = X k E [Fk] /Ck as the traffic load associated with class k and denote by p = C,=l ^pk the total traffic load. It is not directly clear what the right concept of 'load' is in view of the time-vqing transmission rates. In particular. the load as defined above does not coincide with the fraction of time that the base station is active. However, the results in Section IV will show that the abovedefined notion does provide a correct measure of load from a stability perspective.

Assuming packet scheduling results in fair sharing at flow level. the actual transmission rate of the i-th aniving class-k flow: if present at time t_ is:

K

where ^n. denotes the lotal number of flows present at time L.

The function C ( n ) accounts for possible throughput gains from clwnnel-aware scheduling. In particular. the function G(n) with C(l) = 1 is increasing in n and tends lo some finite limit value G' for n ⁱ M. while the ratio G ( n ) / n is decreasing in n.

Remark I : Fair sharing trivially occurs in case of static round-robin scheduling for example, corresponding to G(n)

=

1, but it may also naturally arise in case of channel- a w m scheduling. Specifically. in case R k ; ( t )

=

model reduces to that considered in [SI for the flow-level performance of a weight-based scheduling strategy which assigns weights wk = l / C k to class-k uses. In case the users have statistically identical normalized rate varia- tions Yi: E , . . . at packet level, it may then be shown that G ( n ) = E [max{Yi,. . . ^~ Y;}]. As may funher be deduced from [l]. [PI. [18]. [MI. the Proportional Fair algorithm for the CDMA 1.sEV-DO system would approximately behave like the weight-based strategy. provided the exponential smootlung window is suiliciently large. In case the feasible transmission rate R k i ( L ) is (slowly) timne-van.ing_ similar arguments suggest that a weight-based strategy which assigns a dynamic weight

u i k i (1) = l / R k i ( t ) to the i-th class-k user. results in the actual transmission rate (1) at flow level.

Remark 2: The comparison results to be derived in Sec- tions V and VI in fact remain valid under the even milder

07803-8355-9/01PS20.M) 02034 IEEE. 1873

assumption that the i-th class-k user receives service at rate R k i ( t ) N k ( n i , . . ^{. , n ~ ) .} (2) where 'nk denotes the number of active class-k flows and the function H k ( . ) is decreasing in each of the nk's. Unfortunately however. when the function H k ( . ) is not of the form G ( n ) / n with r~ =

E,"=,

111. DEFINITION OF FLUID A N D QUASI-STATIONARY RECILIES

The flow-level model defined by (1) corresponds to a Processor-Sharing type queue wheere the service rate of each customer is modulated by an independent stochastic process.

Considering the extreme difficult?. of analyzing such a system, we introduce hvo limit regimes, termed puid and quasi- sfafionarv_ where the rate processes evolve on an infinitely fast and a n infinitely slav time scale, respectively. Formally, let us consider a family- of systems. parameterized by s E (0: 03)-

where the generic rate process for class-k flows is R t ) ( t ) ^E Rl;(st). Thus the parameter s represents the 'speed' of the rate process. In case R s ( t ) is a Makov process, the process R t ' ( t ) may be obtained by scaling the transition rates~with s.

When the parameters grows large. the rate process appmx- imately averages out over the time scale of the How dynamics.

In the limit for s

-

-

Accordingly, we define the class-k traffic loads in the fluid and quasi-stationary regimes: pg = XkE [Fk] / C , and p y = XkEIFk/Rk(0)] = X k E [ F k ] / q . where = E[l/Rk(O)]-'. Note that these load factors depend on tbe rate statistics only through the arithmetic and harmonic means.

respectively. By Jensen's inequality, we hn;c p! I p r . Denote finally by pR E p y the toral traffic loads in the fluid and quasi-statiomy regimes. respectively. Recall As mentioned earlier. the fluid and quasi-stationay regimes are particularly relevant, because their performance can be ex- plicitly evaluated. Based on the results of [Z], it can bc shown that a neccssary and sniiicient condition for stabifit?. of the

&i

d (ESP. quasi-stationary) regime is pR < G' (resp. pq" <

G'). In case of stabiliv, the stationary distributions irA and

irqs of the numbers ( n l , . . . ^~ . n K ) of on-going flows of each class in the fluid and quasi-statiormy regimes depend on the tr&c and rate statistics through the loads pg and p y only:

-.

p; and pqs E

that p c p f l .

where ~ ~ ( 0 ) and ^nq5(0) are determined by the nonnalizng condition and n =

Et=,

Alternatively, the performance can be naturally measured in terms ofpow, fhmrighpur:

When G ( n ) regimes. respectively:

1. we obtaiR for the fluid and qnasi-statiormy 7: = G ( l - p f l ) and

TF

IV. ST-ABILITY CONDITION

We say that the system described in Section I1 is stable if.

starting from any initial state. it converges to a finite statiomy regime. It follows from the stochastic bounds to be derived in Section V that the condition pR < G' is necessary for stability, while the condition pq" < G' is sufficient. Note that when the number of on-going Bows tends to oo_ each flow stays a long time in the system so that the rate process tends to average out over the flow duratioh i.e.. the system behaves a s in the fluid regime. Thus we expect the condition p

=

Theorem I: If p < G*, then the system is stable

Remark 3: The assumption of fair sharing is crucial for the above stability condition to hold. This condition may be relaxed by giving priority to those flows with the highest feasible rates. In a high mobility scenario, the BS would then transmit to a user only when shehe is close to the BS, a strategy closely related to that considered in [I31 in the context of ad hoc nemorks. In the present c o n t e a however.

fast variations in the feasible rates are already exploited at the packet level by oppomnistic scheduling and slower variations cannot be exploited without severe impact on user performance due to starvation effects (refer to Section I).

v. COMPARISON WlTlI FLUID AND QUASI-STATIONARY REGIMES

We now compare the performance of the system with that in the fluid and quasi-statio- regimes. using the notion of stochastic ordering (see for instance [21]).

Definirion I: ( s t and icc orderings) Let X and Y' be ^hV0 r.v.'s on W". Write X Ist Y' (resp. X Y) if and only if E [ f ( X ) ] 5 E [ f ( Y ) ] for all increasing (resp. increasing and convex) functions f : W" + I3 for which the previous expectations exist.

Note that these orderings are particularly relevant. since X Y allows thecomparison of the distributions of X and 0-77803-8355-9/04/$20.00 02004 Em. 1874

Y , i.e._ Pr [X 5 z] 2 Pr [Y 5 z] for all T . whereas X

si..

Assume that the system is empty at time 0 and denote by N k ( t ) the number of active class-k flows at time t. For i = 1,. . . , N k ( t ) , let &(t) be the remaining size of active class-k flow i at time t. We define the total workload at time t as:

Theorem 2 below. proved in Appendix B, states that perfor- mance improves (resp. deteriorates) in terms of the number of active flows. the workload and the response time T of an Aitrar?. flow. when the rate processes of some flows satisfs-ing Assumption 1 below are replaced by the corresponding fluid (resp. quasi-stationary) versions as described in Section III.

Assrrmption I: The cumulative distribution function (c.d.f.) P(.) = Pr [F

<

Note that Assumption 1 is satisfied by a broad class of distri- butions. including qmnential. hyper-expnential or Weibull.

In particular, it is possible to represent the highly variable flow size distribution of typical data networks.

Theorem 2: We have, for all k = 1: . . . , K . Wfl(t) < I c z U/(t)

si..

I W t ) ^<,t Ndt) I s t N m ) .

(4) ( 5 ) Tfl 2.t T < a t ps, (6) where the superscript fl (resp. q") refers to the system where the rate processes of some set of flows satisfying Assumption 1:

are replaced by the corresponding fluid (resp. quasi-stationary) processes.

The above comparison results are also valid when the system is in equilibrium. Denote by W(m), N k ( m ) and Tk(cu) the workload, the number of active class-k flows and the response time of class-k flows in steady state. respectively. We deduce the inequalities (8) and (9) in the next corollaly from Theorem 2 and the stability of the st-order by limits [Zl]. The inequality (7) results from (4) and a classical monotonicity properly of the Loynes' constrnction as explained in [j.

page 2811.

Corolloq~ I: Let C c { l > . ... I < } be an a h i t m y subset of classes that satisfs- Assumption 1. We have, for all k = 1,. . . , K :

(7) (8) (9) where the superscript (resp. q") refers to the system where the rate processes of the flows of the classes in C are replaced by the corresponding fluid (ESP. quasi-stationary) processes.

Note that the above fluid and quasi-stationary regimes corre- spond to those defined in Section I11 when C = {l, . . ^{. , K}}

in which case they provide tractable upper and low-er perfor- mance bounds.

I.I'fl(oo)

sicz

VI. IMPACT OF THE SPEED OF RATE VARIATIONS

In this section we investigate how performance varies with the time scale of the rate processes. In order to do so, we suppose that the processes Rk(t) for some users are repiaced by processes R r ) ( t ) Rk(st) for some constant s > 1.

The constant s may be interpreted as an acceleration factor.

Although one might conjecture that performance impmves when the rate process is speeded up, this result does not hold in certain ve? specific cases [22]. However, the monotonity property can be established when the rate process satisfies the following assumption

Assumption 2: The rate process is a homogeneous station- ary Markov process. The transition kernels Q and Qr of the Markov process and of the corresponding time-reversed Markov process are <,,-monotone. Recall that Q is S L - monotone if and only if, for all increasing functions f- the function z U Jf(t)&(z, d t ) is also increasing [21].

Assumption 2 is satisfied by a broad class of processes.

including birthdeath processes and Markov processes with a mscrete state space and a genemtor Q = ( q i j ) such that q i j

does not depend on i [ 5 ] .

The next theorem. proved in Appendix B. states that perfor- mance improves when the rate processes of some set of flows satisfying Assumptions 1 and 2 are accelerated. Speeding up some users (or equivalently their rate process) improves the performance for all flows.

Theorem 3: We have. for all s > 1 and all k = 1 : . ..~li:

CV(3)(t)

si..

N P ) ( t ) I s t A u t ) , (11) T ( " ) Ist T, (12) where the superscript refers to the system where the rate processes of some set of flows satisfying Assumptions 1 and 2, are speeded up by a factor s.

The next corollary presents the connterparl of Corollary 1 Coro//oty 2: Let C c { 1 , . . . , I<} be an arbitmy subset of classes that satisfy Assumptions 1 and 2. We have. for all s > 1 and all k = 1,. . . ^~ K :

VV(")(m) sicz bV(m), (13)

Nf):S)(m) I s t -wk(oo), (14)

%"'(CO) S LZ x X ) , (W

where the superscript ^(') refers to the system where the rate processes of the flows of the classes in C are sped up by a factor s.

0-7803-8355-9/04/$20.00 02004 EEE. 1875

Ring j Rate c j Radius rj (Kbitis) (a = 4)

In the following. we consider a circular cell of external radius R = ^{T L} corresponding to L

+

SINR (dB)

intensity A. The probability p j that a new flow s t a t s its sewice in ring j is proportional to the surface of this ring.

i.e.. p l = (T: - r j - l ) / R 2 , The flow tluoughputs in the limit regimes are given by (3). Simulation results are obtained for exponentially distributed flow sizes of unit mean and Markov rate processes with values in ^{{CO, CI,} . . . , ^CL}. We make the natural assumption that the rate process can only jump between adjacent states. so that for each class. the Markov rate process is a birthdeath process. Note that Assumptions 1 and 2 are satisfied.

.4. Low mobilip

In the low-mobility scenario. the feasible rate of a user typically evolves in a set of 2 to 5 consecutive rates: roughly corresponding to SINR variations of 5 to 12 dB (sec Table I).

Rather than fitting a log-normal distribution. we simply assume that the feasible rate of each user takes a fixed number of values, a_ and that all transitions rates of the corresponding birthdeath process are equal. We performed simulations not reported here to verify that performance depends on shadowing mainly through its amplitude and not on its precise distrihu- tion. We consider a cell of radius R = 1.86 (thus L = 6) and evaluate performance in the following two cases:

.

i, ?: i.

.

andtheirfeasible r a t e s a r e c ~ ~ ~ , ~ ~ - ~ ~ c ~ . c l i + ~ . ~ ! l i + ~ ~ v i t h corresponding marginal probabilities

6: i, i, i, Q.

Figure 1 presents the throughput of flows of classes 1 and 5 as a function of cell load in case of shadowing with low amplitude (a = 3) and with different values of the speed s. In Figure 2, we present the throughput of flows of classes 1 and 3 in case of shadowing with high amplitude ( a = 5).

Fluid regime -

speed = 0.1 1.6 "i

f ^1.4

..

"

0 0.2 0.4 0.6 0.8 1

Tranic load Fip. 1.

curves) in c a e of shadowing of low amplitude (a = 3).

Throughput of Rows of elms 1 (upper curves) and cIm 5 (lower

As expected in view of Corollaries 1 and 2. the fluid and quasi-statiow regimes provide optimistic and conservative estimates of the throughput. respectively. and speeding up the rate processes improves performance. Funher ObSeNe that the

0-7803-8355-9/04/520.00 OZCC4 IEEE. 1876

0 0.2 0.4 0.6 0.8 1 Tranie load

Throughput of flows of clasis 1 (upper C U N ~ S ) and class 3 (lower Fig. 2.

curves) in case of shadowing of high amplitude (a = 5).

- Y

limit regimes only differ significantly in case of shadowing with high amplitude.

E. High mob;/;@

We ^MW consider a high-mobility scenario where the vari- ations in path loss cannot be ncglccted. We assume that a fraction /3 of the usem move across the entire cell whilc the others are static. We do ~t account for shadowing_ i.e..

G,

.

Rh(t) Ck = c k for all t. The load associated with this class is p k = (1 - p ) X p k / C k .

Class-(L+ 1) users move in rings 0 : . . . , L according to a birthdeath process with marginal distribution PO,. . . ^~ p ~ , corresponding to isotropic motion in the cell. so that CL+I = C k p k c k . The load associated with tlus class Figure 3 gives, for a cell of radius R = 2 (thus L = 7) where all usem move ( p = 1). the flow throughput as a function of

1. There are h' = L

+

is p L + 1 = P W L + I .

.~ '+.

.\a,

Speed = 0 . 1 0.8

OS regime ^~~~~~~

0.6

OS regime ^~~~~~~

0.6

0.4

0

0 0.2 0.4 0.6 0.8 1

Trait load

0 0.2 0.4 0.6 0.8 1

Trait load Fix. 3.

in a cell of radius R = 2.

Flow throughput as n function of tmffic load when all u e n move

r

-

0.3 c

9 0.2

r e

s 0.1

.

0s Simulation regime ~~~ . ~~~ ~..

t . f

0.01 0.1 1 10 104

Speed

Fig. 4. Flow throughput as a function of speed for a fixed traffic load p = 0.5 when all wen move in a cell of radius R = 2.

total traflic load for mfferent values of thespeed s. The impact of speed on 00w throughput for fixed load p = 0.5 is shown in Figure 4. Figure 5 is the analog of Figure 3 for a cell of radius R = 1.19 (thus L = 2). Note that for large variations in the feasible rate (see Figure 3). performance is very sensitive to speed. whereas for limited variations (see Figure 5), the

fluid and quasi-stationary bounds are v e y close, indicating

- *

kb

-

\.\

a

that performance is approximately insensitive.

0-7803-8355-9/04/s20.00 Q2W I E E . 1877

Figure 6 gives the flow throughput of static users in ring 0 and of moving users. when the proportions of static and moving users are the same (fl = 0.5).

0 0.2 0.4 0.6 0.8 1

Traliic bad

Fig. 6 .

moving UCR (lower curves) in B cell of radius R = 2.

Flow throughput of static users in ring 0 (upper curves) and of

The numerical results suggest that performance is sensitive to the speed of the fading process only when Tanations in the feasible rates of those users representing a significant part of the total traffic load are of highamplitnde. When the amplitude is lowy performance is almost insensitive. i.e.. essentially depending on the traffic and fading statistics through the traffic load of each class only. In this case. the quasi-stationary regime provides an accurate consewative estimate of flow- level performance.

VIII. CONCLUSION

We have examined the impact of slow. mobility-induced rate variations on the flow-level performance of a wireless data system. We have compared the performance of the system with that in two limit regimes. termed fluid and quasi- stationary. where rate variations occur on an infinitely fast and an infinitely slow time scale. respectively. The fluid and quasi- stationary regimes provide explicit performance estimates, which are provably optimistic and conservative, respectively.

Besides, the performance of the limit regimes is insensitive.

and only depends on an appropriately defined load factor. thus yielding bounds that only involve simple first-order system parameters For a broad class of Makov-type fading processes.

we further proved that performance varies monotonically with the time scale of the rate variations:

At a qualitative level. the finding. that mobility-related rate variations impmve performance resembles the generic principle underlying opportunistic scheduling. In the present context> hoivever. the performance improvement does not rely on opportunistic scheduling. Instead. informally stated.

it arises from the fact that flow-lmel performance measures behave as convex functions of the rate processes.

From a practical perspective, when the traflic load generated by those users with large rate variations is limited. the quasi- s t a t i o ~ w regime provides an accurate conservative estimate of flow-level performance. This allows the development of simple dimensioning d e s os cf users were static, as derived

in [7]. In cases where moving users with large rate variations reptesent a significant part of the total tratfic load. performance becomes sensitive to the precise traflic and fading statistics. It may then be necessary to take mobilit?. and shadowing effects into account.

Note finally that the positive impact of mobility relies here on the assumption of perfect rate predictions. Tlus is reflected in the model by the fact that the inarginal distributions of the feasible rates do not depend on the time scale of rate variations.

It would be very interesting to study the extent to which the estimation and prediction problems due to high-frequency fad- ing countetbalance the performance improvements established in the present paper.

APPENDIX A. Stobiiiiy condition

Pmof of Theorem I: We prove the result for Cox flow size distributions which are known to form a dense subset of the set of all distributions with non-negative support. Specifically! we assume that class-k flows have i.i.d. exponential sizes of mean l / p k > and generate a new class4 Bow with probability pkl

when completed. By creating additional classes and dividing each random flow size into a random number of exponential phases. the model is then sufficiently general to cover a v flow size distribution. The total tratfic load is given by:

p = x ( I - P ) - l ( b & C ) - 1 ,

where X = (A,) is a row vector, I is the identity matrix.

P = ( p k ~ ) , and p = ( @ e ) . C = ( C k ) are diagonal matrices.

Similarly, we assume that the rate process R k t ( t ) is a function of a finite-state Markov process m k l ( t ) . By increasing the number of states, such a Markov process can approximate any stationary and ergodic process.

The stochastic process { N ( t ) , m ( t ) } where N ( t ) and u ( t ) denote the row vectors ( N e ( t ) ) and ( u k i ( t ) ) , k = 1,. . . , I < >

i ₌ 1,. . . ^~ N k ( t ) , respectively, is an irreducible M&ov process. Define the workload at time t as:

W ( t ) = J N ( t ) ( I - P)-1(pC)-lI Assume that p < G' and let t o

sequence of initial states { ~ ( j J ( o ) > u ( J ) ( ~ ) } j with l/(G* - p ) . For any

= 1:

lim ^~

3-00 j

I.V(J) (0)

we will pmve that the sequence of workload processes { I V ( J J ( L ) } , satisfies for any t < t o :

As the workload defines a Lj-apunov function for the Markov process { X ( t ) . o ( t j } , the proof then follows from Foster's stability criterion [20].

Denote by A(tj. B ( t ) and D ( t ) , respectively. the row vec- tors of the number of exogenous arrivals. endogenous amvals and departures of class-k flows up to time t , k = 1.. . . , I C . We have:

N ( t ) = N(0) + A ( t )

+